Lagrangian Particle Methods for Biological Systems - Porous Media: Applications in Biological Systems and Biotechnology - page 257

diffusion from or to the biofilm, while typical “negative” processes include
cell death, erosion (the removal of single cells or small cell clusters from the
biofilm), and fracturing or sloughing (the removal of a large number of living
and/or dead microorganisms in a single event) due to liquid-biofilm hydro-
dynamic interactions. The effects of possible physiological and biochemical
factors were not considered in the current model. However, in principle, those
effects can be incorporated in more comprehensive models.
In the DPD model, we use a mathematical description that is widely
employed in the biofilm literature (Picioreanu et al. 2000; Eberl et al. 2001).
The entire computation domain is divided into three phases, namely the liq-
uid phase, the biofilm phase, and the biofilm support (substratum) phase. In
general, the presence of two types of nutritional substrate, electron donors
and electron acceptors, are required for biofilm growth. If there is an unlim-
ited supply of one substrate (either electron donors or electron acceptors)
then the biofilm growth is limited by the concentration of only one nutri-
tional substrate, S (Picioreanu et al. 2000). Then the biofilm growth, and
the advection and diffusion of nutritional substrate, S , are governed by the
advection-diffusion-reaction equation:

2 C s −

∂C s /∂t + V ·∇ C s = D s ∇ r s (7.1)

where V is the liquid velocity, C s ( x ,t ) is the concentration of substrate, S ,at
position x and time t , and D s is the diffusion coecient of the substrate. The
substrate consumption rate (kg m − 3 s − 1 ) in the biofilm phase, r s , represents
substrate consumption due to biofilm growth. Here we assume that for a single
substrate, r s , the substrate consumption rate is given by a Monod function
(Picioreanu et al. 2000):

r s = µ m
Y bs

C b C s
( K s + C s )

(7.2)

where µ m is the maximum biomass growth rate (s − 1 ), K s is the substrate satu-
ration constant (kg m − 3 ), Y bs is the dimensionless biofilm yield (kg biomass/kg
substrate) for substrate S , and C b ( x ,t ) is the biomass density.
The liquid velocity field, V , is given by the incompressible NS equations

∇· V = 0 (7.3)

2 V

∂ V /∂t + V ·∇ V = −∇ P / ρ + ν ∇ (7.4)

Equations (7.3) and (7.4) describe the mass and momentum conservation
in the liquid phase, where P is the pressure field, ρ is the liquid density, and ν
is the kinematic viscosity. The kinetic equation describing the biofilm growth
and/or decay in the biofilm phase is written as

dC b / dt = Y bs ( r s −

m s C b ) (7.5)

where m s is a maintenance coecient (kg substrate/[kg biomass s]) represent-
ing the biomass simultaneous decay effect. Spreading is an important char-
acteristic in the biofilm kinetics model. The biomass density has a maximum